Mathberry Lane

24
Jul
2024

The Ultimate Math Resource Bundle 2024 is Here!

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17
Jul
2024

Mastering Surface Area: Teaching Tips for Math Teachers

categories: Geometry

Teaching surface area can be one of the more challenging topics in geometry, but with the right strategies and tools, it can become an engaging and rewarding experience for both teachers and students. As a seasoned math teacher, I’ve developed several lessons to help students grasp the concept of surface area. In this blog post, I’ll share these strategies and introduce you to a valuable resource that will make your teaching even more effective: the Surface Area Formulas Graphic Organizer.

Why Starting with a Low Floor Warm Up is Important

Engaging students at the start of the lesson is crucial. If you lose their attention in the first few minutes, it can be difficult to get their attention back. One way I have found to engage all students at the start of a lesson is to incorporate an entry level warm up prompt. I really like using “What do you notice?” and “What do you wonder?” as entry level prompts. All level of students can be successful with this task and I am always surprised when they notice or wonder ideas I would never have thought of myself.

Strategies for Teaching Surface Area

Nets: Start the unit with a hands on introduction lesson to help students better understand what surface area is all about. Exploring different nets is a great activity to help students understand surface area for different shapes.
Vocabulary: When introducing formulas for surface area, it is important to review each vocabulary term that is part of the formula. For example, if the surface area of a sphere has “r” for radius in the formula, then make sure that students know the difference between a diameter and a radius.
Review Basic Area formulas: When I taught surface area to my class, I found that some students needed a review of finding the area of basic 2 dimensional shapes. These shapes would be the bases of our 3 dimensional solids.
Real-Life Applications: Connect the topic to real-life situations. Have students calculate the surface area required to wrap a gift, the leather of a baseball, or the fabric of a tent. These practical applications make learning more relevant and interesting.
Practice, Practice, Practice: Provide ample opportunities for practice. Use a variety of problems so students can apply their knowledge in different contexts.

Need a Surface Area Graphic Organizer?

I’ve got you covered! To further support your teaching, I’ve created a Surface Area Formulas Graphic Organizer, a powerful tool designed to help students visually organize and remember the different surface area formulas. This organizer includes formulas for finding the surface area of the following solids:

Sphere
Cylinder
Prism
Pyramid
Cone

By using this graphic organizer, students can easily reference the formulas and understand the relationships between different shapes. It’s an excellent resource for in-class activities, homework assignments, and test preparation. And, I’ve included 2 pages of practice worksheets too. Grab your free copy below!

Interested in checking out my lessons? If so, visit my TPT shop linked here or click on an image below.

I hope these tips and resources help you in your classroom. Happy teaching!

28
Apr
2024

Surface Area of Spheres Investigation with Oranges

categories: Geometry

I love a good hands on activity that helps my students remember the math they are learning. A great activity to remember the surface area of a sphere formula is to investigate the surface area of an orange.

Make sure to use oranges instead of clementines as the clementines are not as spherical as oranges. Students trace the outline of an orange onto a piece of paper, keeping their pencil perpendicular to the paper as best they can. Then have students use a compass to copy the same size circle several more times on their paper. Ask students to make a conjecture about how many circles they believe the orange peel will cover once they have peeled the orange.

Once students have their circles and conjecture, it is time to peel the orange. Students peel the orange and then test out their conjectures by covering each circle, one at a time. Students should find that the orange peel is able to cover 4 circles. Then as a class, make the connection between the radius of a circle and the radius of a sphere, like an orange. Ask students to write a formula for the surface area of their orange.

If you would like to grab a free copy of my Surface Area Investigation Activity, here is the link:

Alternate Ways to Facilitate this Activity:

Each student completes the activity on their own(an orange for each student)
Students work in groups of 3 or 4 students (7-10 oranges)
Teacher has student volunteers go to the front of the room to model the activity for the class (1 orange)

01
Aug
2023

Teaching Increasing and Decreasing Functions

categories: Key Features of Functions, Mathberry Lane Resources on TPT

For some reason students tend to have difficulty with this concept. At first glance, it seems rather straightforward. If a graph is going up, then the function is increasing. If the graph is going down, then the function is decreasing. If the graph is a horizontal line, then the function is constant. This is the level of this key feature of functions taught in my algebra 2 class.

In calculus, we take it a step further and look at intervals where the first derivative is positive to see where the function is increasing. And likewise, intervals where the first derivative is negative tells us where the function is decreasing. If the first derivative equals zero, then the function is constant.

Next, are the intervals for increasing and decreasing open or closed? Different textbooks will have different notations. Some will use open intervals while others will use closed. I used to be in the open interval camp, but have switched over to closed. I discuss this with my students about how textbook publishers and even math teachers cannot agree on this topic. With that said, if a function is increasing as it goes to an asymptote or infinity, then an open interval would be used.

Let’s look at the definition for a function increasing or decreasing on an interval.

I start this topic with a what do you notice prompt for the graph shown above. It is an entry point into the lesson where each student can share what they see. We then get out some crayons or colored pencils to shade in different portions of the graph.

Next I ask students to try sketching their own examples of graphs that are increasing, decreasing or constant, followed by a graph that exhibits more than one of these behaviors.

Before we start creating intervals for increasing and decreasing, we first review interval notation. Click HERE if you are interested in my free graphic organizer for interval notation.

Finally, we analyze graphs to identify intervals of increasing, decreasing or constant. After students feel comfortable identifying these intervals, I give them intervals with specified criteria and ask them to create the graph that would meet these criteria.

If you are interested in my lesson, it is available in my TPT shop as part of my Key Features of Functions Unit. I have included 2 versions: closed interval notation and open interval notation. Cheers!

30
Jul
2023

Teaching Inverse Functions in Algebra

categories: Uncategorized

In elementary school, students learn how some mathematical operations are opposites of other operations. Addition is the opposite of subtraction. Multiplication is the opposite of division. As students get older, we introduce the word inverse. Inverse operations undo what has been done and are essential for solving equations. In algebra we expand the idea of inverses to functions. In geometry, the inverse of a conditional statement is when the hypothesis and conclusion are negated. In calculus students learn that integration is the inverse of taking the derivative.

Today we are going to delve into inverses with respect to the algebraic lens of functions, both analytically and with the geometric connection graphically.

Graphical Inverse

A function and its inverses are connected graphically by being the reflection over the line y=x.

When I start teaching inverses of functions, I often will start with points on the coordinate plane and have students switch the x and y coordinates to find the inverse. If you do this with several points and then ask students to graph the line y=x, they often will notice the connection of the reflection over the line y=x on their own. Here is a sample prompt to give your students.

We would discuss the domain and range of the function and its inverse as well as deciding if the inverse is still a function or just a relation.

Next, we look at a linear function, because this is the function that my students are most comfortable with and you do not need to restrict the domain for the inverse to be a function.
I may use a function similar to the graph shown in the first image above, f(x)=2x+4. Students will table the function and then switch the x and y coordinates to graph the inverse. They then would graph the line y=x and really see the connection that the function and its inverse are reflected over the line y=x. It would be good to note whether the inverse of the function is also a function at this point, as it will set up a future discussion when graphing parabolas.

After students start to get a feel for what an inverse looks like graphically, it is time to introduce the algorithm for finding the inverse.

Steps to Find the Inverse of a Function Algebraically

We will go over the steps and an example followed by some practice problems.

Restricted Domains

Finally, we will look at functions that need to have a restricted domain to make the function’s inverse also a function. It should be noted that if the inverse is not a function, then we would classify it simply as a relation.

The parent function for quadratics is a great function to start with when looking at restricted domains.

How do you teach inverse functions? Leave a comment below! If you are interested in my lesson or more teaching resources for inverse functions, visit my TPT store below!

02
Jul
2023

Teaching Interval Notation

categories: Key Features of Functions

Interval notation is a way to describe a set of numbers. In elementary school, students are introduced to number lines and inequalities. Number lines are a wonderful visual tool for students to make sense of numbers and to process whether a value is greater than or less than another value. Eventually, these ideas morph into a solution set when they reach pre-algebra. For example, x is greater than 3 has infinitely many solutions. Its solution set could be displayed on a number line with an open ended ray.

Students first learn to write this analytically as an inequality, x > 3. We can write this same interval of values with interval notation such as (3,∞). In algebra, we start to study intervals of numbers such as the domain and range. Instead of using an inequality to represent an interval of real numbers, interval notation is often used instead. When a set of numbers does not include the endpoint, as shown above, a parenthesis is used to indicate that the interval approaches that number, but does not include it in the interval.

Suppose we would like to include the endpoint of an interval. Let’s look at the interval when x is less than or equal to 2.

As an inequality we would write this as x ≤ 2. However as an interval of values, we would use a square bracket to show that the endpoint 2 is included in the interval such as (-∞, 2]. Note that one can never actually reach infinity, so infinity will always have a parenthesis and not a square bracket.

Sometimes an interval of numbers has a starting and ending point on the number line. This is sometimes referred to as an “and” compound inequality.

As an inequality we would write -3<x<1. And then for interval notation it would look like (-3,1).

We can make a union of intervals when the rays go in opposite directions. This would be an “or” compound inequality.

The set of all real numbers can also be written with interval notation.

A common mistake I students make when using interval notation is to write the larger value first and then the smaller value. So, make sure that when you introduce interval notation to your students that you remind them it looks like (lower bound, upper bound).

You can grab my free interval notation graphic organizer here. I also have a full version Interval Notation Lesson available to purchase in my TPT store that is part of my Key Features Unit.

22
Aug
2020

The Importance of Mathematical Modeling

categories: Mathberry Lane Resources on TPT

Many students have a tough time visualizing algebra. One way to help students visualize math is to make a model. Modeling usually starts in the elementary grades.

Mathematical Modeling from numbers to arrays to area models to polynomials

Modeling can start simply by modeling a numerical value with pictures or manipulatives. For example, to model the number five, a student could draw 5 circles, 5 cubes, 5 whatever’s…

As students start learning about larger numbers, they begin to use arrays to model these values. For example, the number 12 might be modeled by an array with 3 rows and 4 columns. This is building the foundation for understanding multiplication. Eventually 3 rows by 4 columns turns into 3×4 =12.

The array modeling of 3×4=12 then leads to area models. A rectangular figure can be labeled with dimensions to represent the side lengths of an area.

You can then use an area model to help students model the distributive property. 5(10+2)

Eventually, we replace a value with x, but we can still use the idea of an area model to help conceptualize the distributive property. Let’s look at 5(x+2).

And finally more complex polynomials can be multiplied such as (x+2)(x+3). I hope that seeing these models helps you understand the importance of the mathematical progression of modeling a basic number like 5, because it helps students conceptualize the abstract later on in algebra.

model multiplying polynomials with algebra tiles

If you are interested in purchasing a resource that has some of these feature, you may want to check out the following resources from my Teachers pay teachers shop.

Mathberry Lane Multiplication Facts — Builds conceptual understanding of multiplication

Mathberry Lane Distributive Property Card Sort — Model the distributive property

Mathberry Lane Multiply Polynomials Task Cards — Also available as Boom Cards or Google Slides Digital Task Cards

01
Sep
2019

How to Study for a High School Math Test

categories: Teaching Tips

Have you ever heard a student ask how to study for a math test? Or maybe, have they told you that they did not study because they don’t know how to study for a math assessment? Some students still do not know how to study in high school and college is only a few years away. The time to teach them how to study is now.

Teaching your students to study can be tough love. One fall I polled my students to learn about their studying techniques. I got responses such as:

Study with a friend
Watch videos on Khan Academy
Look over your notes
Re-do some homework problems
Do the review the teacher provides
I don’t study

I told my students that I would not be providing them with a practice test type of review. Some of them were not very happy with me as this is how some of them had been trained to study for a math test since middle school. I asked how many of them were planning on going to college. Most students indicated that they had a desire to attend college. I informed them that their college professors most likely were not going to give them a practice test before their exam. It was time to learn how to study for a math test. High school is a safe place to try out a new technique. If you fail, you can learn from your mistakes, and then reassess(at least at my school). I do not know if this is the case in many colleges.

Goal as a teacher:

My goal was for each student to develop strategies for how to independently prepare for a math assessment.

How I achieved this goal…

This goal is achieved over a series of assessments. Just like students practice newly learned skills, they also have to practice newly learned study strategies and make it become a habit. #1 below goes with assessment #1, #2 goes with assessment #3, etc. Students can use these strategies in other courses besides math as well.

#1 Teach students how to deep dive a lesson working as a team

Students work collaboratively in study groups so that the task is not so daunting
Each group is assigned a lesson from the unit of study which they will present to the class at the end of the period
Students are given a set amount of time to deep dive into their assigned lesson; students are given guidelines(see end of post if you would like a template) for the deep dive which include learning targets or competencies
Students present their lesson overview to the class
- Posters
- Over-sized white boards
- Designated area on chalkboard or whiteboard
- Teacher sets up a google slides doc and students add their presentation to the document which becomes a study guide shared with the class

#2 Assign students to deep dive each lesson independently

After students have practiced the lesson deep dive in teams, for the next assessment, ask them to try it on their own.
Give them 10-15 minutes of class time to work on a specified lesson
Have students trade papers and review a peer’s work and provide feedback
Assign remaining lessons for students to reflect upon for next class
Collect students’ reflections and provide them with feedback

#3 Repeat the process at least once more with collecting student reflections and providing feedback

You should see growth and more detailed reflections the second time students independently complete this activity. Teacher feedback is essential for students to see what might be missing from their reflections or praise what was awesome!

You can continue this process in class as needed or let students be self-directed in their studying. I remind students that they should be reviewing each lesson and processing what they have learned. This can become a great activity for early finishers – start on your unit reflection. If you would like a free template for student instructions for how to deep dive a lesson, you can download the ppt file HERE. I had this printed as a booklet for the first time we did the activity independently. After that, I just printed the cover sheet and let students create their reflections as they saw fit.

06
Aug
2019

Back to School 2019

categories: Giveaway

It’s that time of year. My daughter can’t wait to meet her third grade teacher and my son, going into 7th grade, is soaking up every last bit of summer before we all go back to school in a few weeks.

Don’t miss out on the TPT BTS Sale! Save up to 25% in my Mathberry Lane TPT Store with code: BTS19

To kick things off, I am giving away a $10 Teachers pay Teachers gift card! Head on over to my facebook page to enter! Have a wonderful school year!

21
Aug
2018

Geometry Constructions Digital Unit now available!

categories: Mathberry Lane Resources on TPT

I hope that you had a wonderful summer as we gear up for back to school! I have joined the site wide one day sale for Back to School for today, August 21, 2018. Please use code: BTSBONUS18 at checkout. I have recently finished my latest digital unit for geometry, the constructions unit. It is includes 5 lessons and assessments. These lessons do include paper and pencil labs for students to traditionally experience constructions. I have also included an assessment and a constructions book with a rubric so that it may be used as an alternative assessment. It is now ready for purchase. I have finished the first 3 lessons of the digital parallel lines unit, which includes proofs with angles and lines. I hope to finish up that unit soon! See my store for details.

If you would like to enter the giveaway for the $10 TPT gift card, please visit my facebook page for details. I will randomly select a winner at 9:00 pm EST tonight!

I hope you have a fabulous school year with your students!

~Mathberry Lane